To determine the greatest common factor (GCF) of the expressions [tex]\( 4k \)[/tex], [tex]\( 18k^4 \)[/tex], and [tex]\( 12 \)[/tex], we need to follow these steps.
1. Identify the constants in each term:
- The constants are the numerical coefficients in front of [tex]\( k \)[/tex]. Here, the constants are 4 (from [tex]\( 4k \)[/tex]), 18 (from [tex]\( 18k^4 \)[/tex]), and 12 (which stands alone).
2. Find the greatest common factor of the constants:
- The greatest common factor (GCF) of 4, 18, and 12 is the largest number that evenly divides each of these numbers.
- We list the factors of each number:
- Factors of 4: 1, 2, 4
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- The common factors are 1 and 2. The largest of these common factors is 2.
3. Identify the variables in each term:
- The terms involving [tex]\( k \)[/tex] are [tex]\( 4k \)[/tex] and [tex]\( 18k^4 \)[/tex]. The term [tex]\( 12 \)[/tex] does not contain [tex]\( k \)[/tex].
4. Find the lowest power of [tex]\( k \)[/tex] among the terms that contain [tex]\( k \)[/tex]:
- [tex]\( 4k \)[/tex] contains [tex]\( k \)[/tex] raised to the power of 1.
- [tex]\( 18k^4 \)[/tex] contains [tex]\( k \)[/tex] raised to the power of 4.
- The lowest power of [tex]\( k \)[/tex] is [tex]\( k^1 \)[/tex] or simply [tex]\( k \)[/tex].
5. Combine the GCF of the constants and the variables:
- The GCF of the numerical coefficients is 2.
- The lowest power of the common variable [tex]\( k \)[/tex] is [tex]\( k \)[/tex].
Therefore, the greatest common factor of the given expressions [tex]\( 4k, 18k^4, \)[/tex] and [tex]\( 12 \)[/tex] is [tex]\( 2k \)[/tex].
The correct answer is [tex]\( 2k \)[/tex].
